1.1 Introduction
Vector Autoregressions (VARs) are among the most popular tools in economic forecasting.
VARs offer great flexibility in modelling the complex dynamic relations among macroe-conomic variables, they are easy to estimate and can be used to generate forecasts at multiple horizons (see e.g. Stock and Watson, 2001). However, as even medium-sized VARs (10-20 variables) have several hundred parameters to estimate, potential over-fitting is an immediate threat to forecast accuracy. The literature has therefore either used VARs with only a handful of variables (Chauvet and Potter,2013;Faust and Wright, 2013), or it has resorted to Bayesian shrinkage methods (Banbura et al.,2010). Such methods include Doan et al.(1984)’s Minnesota prior, which assumes that each variable evolves according to a random walk, andWright (2013)’s democratic steady-state prior, which uses long-run forecasts from an expert survey as prior information for the vector of unconditional means.
We build on Wright (2013)’s work and consider a Bayesian shrinkage approach that additionally exploits the non-sample information in survey nowcasts, i.e. forecasts for the current quarter or month. The idea of our approach is that the variables modeled in the VAR and their corresponding survey nowcasts are likely to depend in a similar way on the lagged dependent variables. To exploit this conjecture, we first augment the vector of dependent variables of the VAR with survey nowcasts and then express our belief of similar dependence on the lagged dependent variables through a Bayesian prior.
The idea is best illustrated with a simple example: Consider a variableyt, modeled as a univariate autoregression (AR) with a single lag, i.e. yt=ayt−1+εt, and its nowcast st.
and the prior distribution favoring pairwise identical coefficients can be stated as p way. Through v∆ we express our confidence in this conjecture. If the dependence of the survey nowcasts (st) on the lagged dependent variables is indeed not to dissimilar from the actuals (yt), i.e. if ∆ is small, then the extra information provided through the survey nowcasts will help us pin down the parameters of the original VAR. Put differently, the shrinkage method is likely to reduce the risk of over-fitting the model to the data and we therefore expect it to provide us with more accurate forecasts.1 Note that our
interpretation of shrinkage differs somewhat from the above mentioned approaches: The Minnesota prior, for example, shrinks the coefficients of a vector autoregression towards a system of univariate random walks. Thus, shrinkage is directly provided through the prior.
In our case, instead, shrinkage comes both from the prior and from the data: We shrink one set of unknown parameters (regression of survey nowcasts on lagged dependent variables) towards a second set of unknown parameters (regression coefficients of the VAR). Thus, the method relies on ’learning’ about the parameters of the original vector autoregression from survey nowcasts.
In a forecasting application with U.S. macroeconomic and macro-financial data, we find that a ten-variable VAR estimated using our novel shrinkage approach produces forecasts that are superior to a range of benchmark methods. Specifically, we find that mean squared forecast errors (MSFEs) are typically lower with our method than with a univariate AR(1) estimated by OLS, uniformly lower than with the same VAR estimated using only the Minnesota prior, and comparable to those of survey forecasts.
The idea of similar dependence on the lagged dependent variables can be motivated in several ways: First, empirically, survey nowcasts have often been found to be very accurate predictions of the target variable (e.g. Faust and Wright, 2013). We would therefore expect that they exploit the available information in a way that resembles the true data generating process. Second, Online Appendix B.1 shows that the shrinkage target ∆ = 0 can alternatively be motivated from assumptions about the expectations formation process and about the time series model specification. Specifically, if (i) average expectations are formed in a fully rational manner based on an information set that includes the lagged dependent variables of the VAR, and (ii) the VAR is correctly specified, then the true value of ∆ is zero. The fact that these ideal conditions are not likely to be fully satisfied in practice is one motivation to use ∆ = 0 as a shrinkage target instead of imposing it deterministically.
Similar approaches to incorporate information from survey nowcasts have been used in the frequentist estimation of a three-factor affine Gaussian model for U.S. Treasury yields by Kim and Orphanides (2012), and in the Bayesian estimation of a DSGE model by Del Negro and Schorfheide (2013). However, besides the different model class, a major difference is that these studies have assumed that coefficients are exactly equal for each pair of actuals and nowcasts. By avoiding to impose equal coefficients deterministically, our Bayesian shrinkage method reduces the risk of deteriorating forecasts by imposing restrictions that may turn out to be severely erroneous.
Recently, a number of studies have used exponential tilting (Robertson et al., 2005)
(in terms of mean squared error (MSE)) asymptotically translate into superior forecasts (in terms of MSFE).
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1.1. INTRODUCTION to incorporate moment restrictions - for example from survey forecasts - into predictive densities obtained from macroeconomic time series models. Exponential tilting proceeds in the following way: From the universe of densities fulfilling the moment restrictions, it chooses the one closest in terms of relative entropy to the predictive density obtained from the time series model. Using this method,Cogley et al. (2005) have considered forecasting UK inflation with moment restrictions for the mean and variance taken from fan charts of the Bank of England. Alternatively,Altavilla et al. (2014) have used survey point forecasts of short-term interest rates to adjust the forecasts of a Dynamic Nelson-Siegel model of the U.S. yield curve. Lately,Kr¨uger et al. (2015) have employed moment restrictions which represent the mean and variance of survey nowcasts in order to modify the forecast density of a Bayesian VAR. Incorporating survey-based information through exponential tilting differs in a number of ways from our approach: First, it only exploits the survey data after model estimation. Thus, although such information is deemed informative, it is not used to learn about the data generating process but only to modify the forecast density. Second, the method makes no attempt to evaluate empirically whether the moment restrictions (obtained from survey forecasts) it imposes are likely to hold in the data. Exponential tilting therefore relies heavily on carefully selecting the ’right’ moment restrictions. Our method instead lets the data decide how informative survey forecasts are about the data generating process. Eventually, exponential tilting is forecast-horizon specific, i.e. it can only be used to adjust forecasts at horizons for which moment restrictions are available.
By contrast, in our method, survey nowcasts are used to shrink coefficients of a time series model that can provide forecasts at any horizon.
Ba¸st¨urk et al. (2014) present another approach of incorporating survey data into a forecasting model. Specifically, they estimate a new Keynesian Phillips Curve model, using inflation expectations to facilitate estimation of the expectations mechanism. A major difference to our approach is that they effectively include survey forecasts as a regressor, whereas we model survey nowcasts as a by-product of the data generating process. Additionally, while their method is tailor-made for inflation forecasting, ours can in principle be applied to any macroeconomic variable.
The paper proceeds as follows. Chapter 1.2 introduces the methodology and the underlying econometric ideas. Chapter3.4 presents our empirical findings and chapter 3.5 summarizes our results.